3 6 Perpendiculars and Distance Ms Andrejko Real

3 -6 Perpendiculars and Distance Ms. Andrejko

Real World

Vocabulary �Equidistant- the distance between two lines measured along a perpendicular line to the lines is always the same �The distance between a line and a point not on the line is the length of the segment perpendicular to the line from the point. �The distance between two parallel lines is the perpendicular distance between one of the lines and any point on the other line.

Theorems & Postulates � 3. 6 �Theorem � 3. 9 – Two Lines Equidistant from a Third- in a plane, if two lines are equidistant from a third line, then the two lines are parallel to each other.

Examples �Construct the segment that represents the distance indicated.

Examples �Construct the segment that represents the distance indicated.

Steps (when given line and pt. ) � 1. Find perpendicular slope � 2. Make equation using perpendicular slope and given point (P) � 3. Solve system of equations to find point of intersection � 4. Plug into distance formula

Examples �Find the distance from P to ℓ. 1. Line ℓ y=-x-2. Point P has coordinates (2, 4) Perpendicular slope: m=1 Equation through point: y-4 = 1(x-2) y-4 = x-2 y=x+2 -x-2=x+2 y=x+2 System of equations: -2=2 x+2 y=-2+2 y=0 -4=2 x (-2, 0) -2=x Distance formula: √(-2 -2)2+(0 -4)2 √ 32 = 4√ 2 √(-4)2+(-4)2 √ 16+16

Practice �Find the distance from P to ℓ. 1. Line ℓ y=-x. Point P has coordinates (1, 5) Perpendicular slope: m=1 Equation through point: -x = x+4 System of equations: -2 x=4 x=-2 Distance formula: y=-2+4 y=2 (-2, 2) √(-2 -1)2+(2 -5)2 √(-3)2+(-3)2 √ 9+9 y-5 = 1(x-1) y-5 = x-1 y= x+4 √ 18 = 3√ 2

Example �Find the distance from P to ℓ. 1. Line ℓ y=(4/3)x-2. Point P has coordinates (-1, 5) Perpendicular slope: System of equations: Distance formula: Equation through point:

Practice �Find the distance from P to ℓ. 1. Line ℓ y=-3 x+8. Point P has coordinates (-1, 1) Perpendicular slope: System of equations: Distance formula: Equation through point:

Steps (when given 2 parallel equations) � 1. Find y-intercept of line m � 2. Write equation of perpendicular line through yintercept (line p). � 3. Use systems of equations to determine point of intersection between l and p. � 4. Use distance formula

Examples �Find the distance between each pair of parallel lines with the given equations. y=5 x-22 y=5 x+4

Practice �Find the distance between each pair of parallel lines with the given equations. y=-3 x+3 y=-3 x-17